In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected one-dimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal network, and observe that such network carries a multiplicity that in principle can be strictly larger than one. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when the total length becomes large.

Figure 1.
Approximation of globally optimal reinforcement structures for $ m = 0.5 $, $ L = 1, \, 2 $ and $ 3 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture

Figure 3.
Approximation of globaly optimal reinforcement structures for $ m = 0.5 $, $ L = 4, \, 5 $ and $ 6 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture

Figure 2.
Approximation of globaly optimal reinforcement strucutres for m = 0.5, L = 1.5, 2.5 and 5 for a source consisting of two dirac masses. The upper colorbar is related to the weights θ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture